Two men, John and Joseph, can paint a house in 12 days, John alone can paint the house in 20 days. How long will it take Joseph to paint the house alone?
Another way, using ratios:
John paints the house in 20 days, so paints $\displaystyle \frac{1}{20}$ of the house every day.
Joseph paints the house in $\displaystyle x$ days, so paints $\displaystyle \frac{1}{x}$ of the house every day.
Together, every day they would paint $\displaystyle \frac{1}{20} + \frac{1}{x}$ of the house every day.
So if we set up a ratio "amount : days"
$\displaystyle \frac{1}{20} + \frac{1}{x} : 1$
$\displaystyle \frac{x + 20}{20x} : 1$
$\displaystyle x + 20 : 20x$
$\displaystyle 1 : \frac{20x}{x + 20}$.
So that means they would take $\displaystyle \frac{20x}{x + 20}$ days to paint the entire house. Since we are told that together they take 12 days, that means
$\displaystyle \frac{20x}{x + 20} = 12$
$\displaystyle 20x = 12(x + 20)$
$\displaystyle 20x = 12x + 240$
$\displaystyle 8x = 240$
$\displaystyle x = 30$.
So it will take Joseph 30 days.